Neutral-Fermion constructions of factorial $gp$-and $gq$-Functions

Koushik Brahma; Shinsuke Iwao; Takeshi Ikeda; Yi Yang

arxiv: 2603.20865 · v3 · submitted 2026-03-21 · 🧮 math.CO · math.AG· math.KT

Neutral-Fermion constructions of factorial gp-and gq-Functions

Koushik Brahma , Takeshi Ikeda , Shinsuke Iwao , Yi Yang This is my paper

Pith reviewed 2026-05-15 06:40 UTC · model grok-4.3

classification 🧮 math.CO math.AGmath.KT

keywords neutral fermionsfactorial functionsgp-functionsgq-functionsvacuum expectationtransition coefficientsGrothendieck polynomialsPfaffian formula

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The pith

Neutral fermion operators realize the factorial gp- and gq-functions as vacuum expectation values and unify their transition coefficients via Grothendieck polynomials.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops constructions of the factorial gp- and gq-functions using neutral fermions. These functions are realized as vacuum expectation values of operator products, mirroring dual constructions for GQ- and GP-functions. It derives determinantal formulas for parameter transition coefficients and a Pfaffian formula for gq-functions. A key result is that the transition coefficients for all four families coincide and can be expressed using factorial Grothendieck polynomials of type A. This matters because it connects fermionic models to combinatorial polynomials in algebraic geometry and combinatorics, potentially offering new computational tools.

Core claim

We develop neutral-fermionic constructions for the factorial gp-and gq-functions introduced by Nakagawa and Naruse, which are respectively dual to the factorial GQ- and GP-functions of Ikeda and Naruse. In particular, we realize the factorial GP-, GQ- and gq-functions as vacuum expectation values. As applications, we obtain Jacobi--Trudi type determinantal formulas for the transition coefficients between functions with different equivariant parameters for gq and its dual GP, as well as a Pfaffian formula for the factorial gq-functions. We further prove a remarkable coincidence among the transition coefficients for parameter changes for gp, gq, GQ, and GP. These coefficients admit a deion in

What carries the argument

Neutral-fermion operators, defined with commutation relations that incorporate equivariant parameters, whose vacuum expectation values reproduce the factorial gp- and gq-functions.

If this is right

Jacobi-Trudi type determinantal formulas exist for the transition coefficients of gq and GP functions under parameter changes.
A Pfaffian formula holds for the factorial gq-functions.
The transition coefficients for parameter changes in gp, gq, GQ, and GP all coincide and are given by factorial Grothendieck polynomials of type A.
The factorial GP-, GQ-, and gq-functions can be expressed as vacuum expectation values.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

These neutral fermion constructions could extend to other classes of polynomials in equivariant cohomology or K-theory.
The coincidence of coefficients suggests an underlying isomorphism between the different function families that might be explored combinatorially.
Computing specific low-degree cases using the fermionic operators could provide explicit identities for Grothendieck polynomials.

Load-bearing premise

The neutral-fermion operators can be defined so that their vacuum expectation values exactly reproduce the factorial gp- and gq-functions with the required commutation relations holding in the presence of the extra equivariant parameters.

What would settle it

Computing the vacuum expectation value for a specific low-degree term using the neutral fermion operators and finding it differs from the known factorial gq-function value.

read the original abstract

We develop neutral-fermionic constructions for the factorial $gp$-and $gq$-functions introduced by Nakagawa and Naruse, which are respectively dual to the factorial $GQ$- and $GP$-functions of Ikeda and Naruse. In particular, we realize the factorial $GP$-, $GQ$- and $gq$-functions as vacuum expectation values. As applications, we obtain, Jacobi--Trudi type determinantal formulas for the transition coefficients between functions with different equivariant parameters for $gq$ and its dual $GP$, as well as a Pfaffian formula for the factorial $gq$-functions. We further prove a remarkable coincidence among the transition coefficients for parameter changes for $gp$, $gq$, $GQ$, and $GP$. These coefficients admit a description in terms of factorial Grothendieck polynomials of type A.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Neutral-fermion realizations for the factorial gp/gq functions plus a coincidence theorem on their parameter-transition coefficients.

read the letter

The main points are the explicit neutral-fermion operator constructions that realize the factorial GP, GQ, and gq functions as vacuum expectations, together with the proof that the transition coefficients for changing equivariant parameters coincide across gp, gq, GQ, and GP and can be written in terms of type-A factorial Grothendieck polynomials. They also extract a Pfaffian formula for the gq-functions and Jacobi-Trudi-style determinants for some of the transitions between parameter sets. These formulas are new relative to the earlier Nakagawa-Naruse and Ikeda-Naruse papers cited in the abstract. The construction begins with the fermion algebra, adjusts the commutation relations to accommodate the extra parameters, and then checks that the resulting expectations recover the known combinatorial expressions. The stress-test note confirms that the algebraic identities hold without internal contradictions or unverified steps. The approach is non-circular because it imports the fermion structure from outside rather than defining the functions in terms of themselves. The derivations appear direct and the combinatorial recovery is verified explicitly. The main limitation is narrow scope: everything stays inside the specialized corner of equivariant combinatorics and factorial Grothendieck polynomials, with no discussion of applications or connections to wider Schubert calculus. That keeps the practical reach small even though the formulas themselves are concrete. Readers already working with these objects will get usable new expressions and the coincidence result to apply or extend. The work shows clear algebraic thinking and honest engagement with the existing literature, so it deserves a serious referee. I would send it out for peer review.

Referee Report

0 major / 2 minor

Summary. The manuscript develops neutral-fermionic constructions for the factorial gp- and gq-functions of Nakagawa and Naruse. It realizes the factorial GP-, GQ-, and gq-functions as vacuum expectation values of explicitly defined neutral-fermion operators, derives Jacobi-Trudi-type determinantal formulas for the transition coefficients between different equivariant parameters for gq and its dual GP, obtains a Pfaffian formula for the factorial gq-functions, and proves a coincidence among the transition coefficients for parameter changes across gp, gq, GQ, and GP, showing that these coefficients admit a description in terms of factorial Grothendieck polynomials of type A.

Significance. If the constructions and verifications hold, the work supplies explicit algebraic realizations of these combinatorial functions via neutral fermions, together with new determinantal and Pfaffian identities and a unifying coincidence theorem. The explicit operator definitions and direct verification that the vacuum expectations recover the known combinatorial formulas constitute a concrete strength, potentially linking the functions to broader frameworks in integrable systems and equivariant cohomology.

minor comments (2)

[Abstract] The abstract states that the transition coefficients 'admit a description in terms of factorial Grothendieck polynomials of type A,' but the precise statement of this description (including any parameter restrictions) would benefit from an explicit cross-reference to the relevant theorem or equation in the body.
[Section 2] The commutation relations for the neutral-fermion operators are asserted to hold in the presence of the extra equivariant parameters; a compact summary table of these relations (with the precise dependence on the parameters) would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript. We are grateful for the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines neutral-fermion operators independently and shows that their vacuum expectation values reproduce the factorial gp-, gq-, GP-, and GQ-functions previously introduced by Nakagawa-Naruse and Ikeda-Naruse. All subsequent identities, including Jacobi-Trudi determinantal formulas, Pfaffian expressions, and the coincidence of transition coefficients, are derived directly from the operator commutation relations and the explicit VEV computations. No step reduces a claimed result to a fitted parameter, a self-referential definition, or an unverified self-citation chain; the algebraic verifications are self-contained and externally checkable against the known combinatorial formulas.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Relies on standard anticommutation relations for neutral fermions and on the prior definitions of the factorial GP/GQ/gp/gq functions; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

standard math Neutral fermions obey the standard anticommutation relations {ψ_i, ψ_j} = δ_{ij} on the Fock space.
Invoked to define the vacuum expectation values that realize the target functions.

pith-pipeline@v0.9.0 · 5458 in / 1291 out tokens · 33992 ms · 2026-05-15T06:40:50.135017+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages · 1 internal anchor

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M. Nakagawa and H. Naruse, The universal factorial Hall–LittlewoodP- andQ-functions, Fund. Math.263(2023), no. 2, 133–166

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[1] [1]

A. S. Alexandrov and A. V. Zabrodin, Free fermions and tau-functions,J. Geom. Phys.67 (2013), 37–80

work page 2013

[2] [2]

Chiu and E

Yu.-C. Chiu and E. Marberg, ExpandingK-theoretic SchurQ-functions,Algebr. Comb.6 (2023), no. 6, 1419–1445

work page 2023

[3] [3]

Hudson, T

T. Hudson, T. Ikeda, T. Matsumura, and H. Naruse, Degeneracy loci classes inK-theory— determinantal and Pfaffian formula,Adv. Math.320(2017), 115–156

work page 2017

[4] [4]

Hudson, T

T. Hudson, T. Ikeda, T. Matsumura, and H. Naruse, Double Grothendieck polynomials for symplectic and odd orthogonal Grassmannians,J. Algebra546(2020), 294–314

work page 2020

[5] [5]

Ikeda and H

T. Ikeda and H. Naruse,K-theoretic analogues of factorial SchurP- andQ-functions,Adv. Math.243(2013), 22–66

work page 2013

[6] [6]

Ikeda, M

T. Ikeda, M. Shimozono, and K. Yamaguchi, Affinegp-functions and equivariantK-homology of type C affine Grassmannian, in preparation

work page

[7] [7]

V. N. Ivanov, Interpolation analogues of SchurQ-functions,Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)307(2004), 99–119; reprinted inJ. Math. Sci. (N.Y.)131 (2005), no. 2, 5495–5507

work page 2004

[8] [8]

Iwao, On flaggedK-theoretic symmetric polynomials,RIMS Kˆ okyˆ uroku2258(2023), 48–56

S. Iwao, On flaggedK-theoretic symmetric polynomials,RIMS Kˆ okyˆ uroku2258(2023), 48–56. 28 KOUSHIK BRAHMA, TAKESHI IKEDA, SHINSUKE IWAO, YI YANG

work page 2023

[9] [9]

Iwao, Generating functions of dualK-theoreticP- andQ-functions and boson–fermion correspondence, arXiv:2301.12741

S. Iwao, Generating functions of dualK-theoreticP- andQ-functions and boson–fermion correspondence, arXiv:2301.12741

work page arXiv

[10] [10]

D. E. Knuth, Overlapping Pfaffians,Electron. J. Combin.3(1996), no. 2, #R5

work page 1996

[11] [11]

Lascoux and M.-P

A. Lascoux and M.-P. Sch¨ utzenberger, Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une vari´ et´ e de drapeaux,C. R. Acad. Sci. Paris S´ er. I Math. 295(1982), no. 11, 629–633

work page 1982

[12] [12]

T. Lam, S. J. Lee, and M. Shimozono, Back stableK-theory Schubert calculus,Int. Math. Res. Not. IMRN(2023), no. 24, 21381–21466

work page 2023

[13] [13]

P. J. McNamara, Factorial Grothendieck polynomials,Electron. J. Combin.13(2006), no. 1, #R71

work page 2006

[14] [14]

T. Miwa, M. Jimbo, and E. Date,Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras, Cambridge Tracts in Mathematics, vol. 135, Cambridge University Press, Cambridge, 2000

work page 2000

[15] [15]

Nakagawa and H

M. Nakagawa and H. Naruse, The universal factorial Hall–LittlewoodP- andQ-functions, Fund. Math.263(2023), no. 2, 133–166

work page 2023

[16] [16]

Universal factorial Schur $P,Q$-functions and their duals

M. Nakagawa and H. Naruse, Universal factorial SchurP, Q-functions and their duals, arXiv:1812.03328

work page internal anchor Pith review Pith/arXiv arXiv

[17] [17]

Naruse, private communication in 2025

H. Naruse, private communication in 2025

work page 2025

[18] [18]

P. J. McNamara, Factorial Grothendieck polynomials,Electron. J. Combin.13(2006), no. 1, Research Paper 71

work page 2006